Approximating Low-Dimensional Coverage Problems

TL;DR

This paper develops a fixed-parameter tractable approximation scheme for the maximum coverage problem restricted to set systems with bounded VC-dimension, and explores the limitations of greedy algorithms in special cases.

Contribution

It introduces a new FPT approximation scheme for coverage problems with bounded VC-dimension and analyzes the limitations of greedy algorithms in higher-dimensional cases.

Findings

FPT approximation scheme with runtime depending exponentially on (kd/ε)^c

Greedy algorithm's approximation ratio can be improved in certain cases

Greedy and local search algorithms cannot surpass 1-1/e approximation in 4D halfspaces

Abstract

We study the complexity of the maximum coverage problem, restricted to set systems of bounded VC-dimension. Our main result is a fixed-parameter tractable approximation scheme: an algorithm that outputs a $(1-\eps)$-approximation to the maximum-cardinality union of $k$ sets, in running time $O(f(\eps,k,d)\cdot poly(n))$ where $n$ is the problem size, $d$ is the VC-dimension of the set system, and $f(\eps,k,d)$ is exponential in $(kd/\eps)^c$ for some constant $c$. We complement this positive result by showing that the function $f(\eps,k,d)$ in the running-time bound cannot be replaced by a function depending only on $(\eps,d)$ or on $(k,d)$, under standard complexity assumptions. We also present an improved upper bound on the approximation ratio of the greedy algorithm in special cases of the problem, including when the sets have bounded cardinality and when they are two-dimensional halfspaces. Complementing these positive results, we show that when the sets are four-dimensional halfspaces neither the greedy algorithm nor local search is capable of improving the worst-case approximation ratio of $1-1/e$ that the greedy algorithm achieves on arbitrary instances of maximum coverage.